Transcript Solving Equations

Solving Equations
Vocabulary
An algebraic equation is an equation that includes one or
more variables.
An equivalent equations is an equation that have the same
solution(s).
Isolate – to solve an equation you must isolate the variable
(i.e. get the variable alone to one side of the equation).
An inverse operation undoes another operation by
performing the opposite operation (i.e subtraction is the
inverse operation of addition)
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Properties of Equality –
producing equivalent equations
Addition Property of Equality: Adding the same number to
each side of an equation produces an equivalent equation.
x–3=2
x–3+3=2+3
Subtraction Property of Equality: Subtracting the same
number to each side of an equation produces an
equivalent equation.
x+3=2
x+3-3=2-3
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Properties of Equality –
producing equivalent equations
Multiplication Property of Equality: Adding the same number to
each side of an equation produces an equivalent equation.
𝑥
3
𝑥
3
=2
∙3=2∙3
Division Property of Equality: Dividing the same number to each
side of an equation produces an equivalent equation.
5x = 20
5𝑥
𝟓
=
20
𝟓
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Solving One-Step Equations
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Solving an One-Step Equation
Solving an equation with subtraction
Solving an equation with addition
x + 13 = 27
-7 = b - 3
isolate the variable
x + 13 – 13 = 27 – 13 undo addition by
subtracting the same
number from each side
-7 + 3 = b – 3 + 3
undo subtraction by
adding the same number
from each side
x = 14
-4 = b
Simplify each side of the
equation
isolate the variable
Simplify each side of the
equation
Check it!
Check it!
x + 13 = 27
14 + 13 = 27
27 = 27
Substitute the answer into
original equation to check
it.
-7 = b - 3
-7 = -4 - 3
-7 = -7
Substitute the answer into
original equation to check
it.
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Solving an One-Step Equation
Solving an equation with
multiplication
Solving an equation with division
4x = 28
isolate the variable
4𝑥
4
undo multiplication by
dividing the same number
from each side
𝒙
𝟒
𝒙
𝟒
Simplify each side of the
equation
x = -36
=
28
4
x=7
Check it!
4x = 28
4 ∙ 7 = 28
28 = 28
Substitute the answer into
original equation to check
it.
= -9
isolate the variable
∙ 𝟒 = -9 ∙ 4
undo division by
multiplying the same
number from each side
Check it!
𝒙
= -9
𝟒
−𝟑𝟔
= -9
𝟒
-9 = -9
Simplify each side of the
equation
Substitute the answer into
original equation to check
it.
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Solving an One-Step Equation
Solving an equation using reciprocal
𝟒
𝒎
𝟓
isolate the variable
= 28
𝟓 𝟒
5
𝒎 = (28)
𝟒 𝟓
4
m = 35
Multiply each side by 5/4, the
reciprocal of 4/5
Simplify each side of the equation
Check it!
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Solving Two and Multi -Step
Equations
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Understanding
Two-Step
To solve two-step equations, you use the properties of
equality and inverse operations to form a series of simpler
equivalent equations. You can use the properties of equality
repeatedly to isolate the variable.
Multi-Step
To solve two-step equations, you use the properties of
equality, inverse operations, and properties of real numbers
to form a series of simpler equivalent equations. You use the
properties until you isolate the variable.
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You can undo the operations in the reverse
order of the order of operations.
2x + 3 = 15
2x + 3 – 3 = 15 – 3
subtract 3 from each side
2x = 12
simplify
2𝑥
2
Divide each side by 2
=
12
2
x=6
Simplify
Check
2x + 3 = 15
2(6) + 3 = 15
15 = 15
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When one side of an equation is a fraction with more than
one term in the numerator, you can still undo division by
multiplying each side by the denominator.
𝑥 −7
3
3
= −12
𝑥 −7
3
= 3 (−12)
Multiply each side by 3
x – 7 = -36
Simplify
x – 7 + 7 = -36 + 7
Add 7 to each side
x = -29
Simplify
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Using Deductive Reasoning
When you use deductive reasoning, you must state your steps and your
reason for each step using properties, definitions, or rules.
For example:
Steps
Reasons
-t + 8 = 3
Original equation
-t + 8 - 8 = 3 – 8
Subtraction Property of Equality
-t = -5
Use subtraction to simplify
-1t = -5
−𝟏𝒕
−𝟓
=
−𝟏
−𝟏
t=5
Multiplicative Property of -1
Division Property of Equality
Use division to simplify
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Solving a Equation by Combining Like Terms
5 = 5m – 23 + 2m
Original Equation
5 = 5m + 2m – 23
Commutative Property of Addition
5 = 7m – 23
Combine like terms
5 + 23 = 7m – 23 + 23
Add 23 to each side
28 = 7m
𝟐𝟖 𝟕𝒎
=
𝟕
𝟕
4=m
Simplify
Divide each side by 7
Simplify
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Solving an Equation Using the Distributive Property
Make the equation easier by removing the grouping symbols first!
-8(2x – 1) = 36
Original Equation
-16x + 8 = 36
Distributive Property
-16 +8 -8 = 36 – 8
Subtract 8 from both sides
-16x = 28
−𝟏𝟔𝒙
𝟐𝟖
=
−𝟏𝟔
−𝟏𝟔
𝟕
x=−
𝟒
Simplify
Divide each side by -16
Simplify
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Solving an Equation That Contains Fractions
You can use different methods to solve equations that contain fractions.
Method 1
Method 2
Write the like terms by using common denominator
and solve.
Clear the fraction from the equation
𝟑𝒙 𝒙
− = 𝟏𝟎
𝟒 𝟑
𝟑
𝟏
𝒙 − 𝒙 = 𝟏𝟎
𝟒
𝟑
𝟗
𝟒
𝒙−
𝒙 = 𝟏𝟎
𝟏𝟐
𝟏𝟐
Original equation
𝟑𝒙 𝒙
− = 𝟏𝟎
𝟒 𝟑
𝟑𝒙 𝒙
𝟏𝟐
−
= 𝟏𝟐(𝟏𝟎)
𝟒 𝟑
Original equation
Distributive property
𝟓
𝒙 = 𝟏𝟎
𝟏𝟐
𝟏𝟐 𝟓
𝟏𝟐
𝒙 =
𝟏𝟎
𝟓 𝟏𝟐
𝟓
Combine like terms
𝟑𝒙
𝒙
𝟏𝟐
− 𝟏𝟐
= 𝟏𝟐(𝟏𝟎)
𝟒
𝟑
9x – 4x = 120
Multiply
5x = 120
Combine like terms
x = 24
x = 24
Simplify
Divide each side by
16
5 and simplify.
Rewrite the fraction
Write the fraction using
the common
denominator 12.
Multiply each side by
the reciprocal
𝟓
of𝟏𝟐
𝟏𝟐
,
𝟓
Multiply each side by
a common
denominator, 12
Solving an Equation That Contains Decimals
You can clear decimals from an equation by multiplying by a power of 10. First,
find the greatest number of digits to the right of any decimal point, and then
multiply by 10 raised to that power.
3.5 – 0.02x = 1.24
Original equation
100(3.5 – 0.02x) = 100(1.24)
Multiply each side by 103, or 100
350 – 2x = 124
Distributive Property
350 – 2x – 350 = 124 – 350
Subtract 350 from each side
-2x = -226
−𝟐𝒙
−𝟐𝟐𝟔
=
−𝟐
−𝟐
x = 113
Simplify
Divide each side by -2
Simplify
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Solving Equations with Variables
on Both Sides
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How To Get Started
There are variables terms on both sides of the equation.
Decide which variable term to add or subtract to get the
variable on one side only.
To solve equations with variables on both sides, you can
use the properties of equality and inverse operations to
write a series of simpler equivalent equations.
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Solving an Equation with Variables on Both Sides
5x + 2 = 2x + 14
Original equation
5x + 2 – 2x = 2x +14 – 2x
Subtract 2x from each side
3x + 2 = 14
Simplify
3x + 2 – 2 = 14
Subtract 2 from each side
3x = 12
Simplify
𝟑𝒙 𝟏𝟐
=
𝟑
𝟑
Divide each side by 3
x=4
Simplify
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Solving an Equation With Grouping Symbols
2(5x – 1) = 3(x + 11)
Original equation
10x – 2 = 3x + 33
Distributive Property
10x – 2 – 3x = 3x + 33 – 3x
Subtract 3x from each side
7x – 2 = 33
Simplify
7x – 2 + 2 = 33 + 2
Add 2 to each side
7x = 35
𝟕𝒙 𝟑𝟓
=
𝟕
𝟕
x=5
Simplify
Divide each side by 7
Simplify
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Identities and Equations with No Solutions
An equation that is true for every possible value of
An equation has no solution if there is no value of the
the variable is an identity.
variable that make the equation true. The equation
For example: x + 1 = x + 1 is an identity.
x +1 = x +2 has no solution.
10x + 12 = 2(5x + 6)
Original equation
9m – 4 = -3m + 5 + 12m
Original equation
10x + 12 = 10x + 12
The Distributive Property
9m – 4 = 9m + 5
Combine like terms
9m – 4 – 9m = 9m + 5 – 9m
Subtract 9m from each side
-4 = 5
Simplify
Because 10x + 12 = 10x + 12 is always true,
there are infinitely many solutions of the
equation (x can equal any thing and it still will
remain true). The original equation is an
identity.
Because -4 ≠ 5, the original equation has no solution
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When you solve an equation, you use reasoning to select properties of
equality that produce simpler equivalent equations until you find a
solution. The steps below provide a general guideline for solving
equations.
Concept Summary
Solving Equations
Step 1
Use the Distributive Property to remove any grouping symbols. Use properties of
equality to clear decimals and fractions.
Step 2
Combine like terms on each side of the equations.
Step 3
Use the properties of equality to get the variables terms on one side of the
equation and the constants on the other.
Step 4
Use the properties of equality to solve for the variable.
Step 5
Check your solution in the original equation
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Do You Know How?
Solve each equation. Check your answer.
Solving one-step equations
Solving two-step equations
1. x + 7 = 3
5. 5x + 12 = -13
2. 9 = m – 4
6. 6 =
3. 5y = 24
4. You have already read 117 pages of a
book. You are one third of the way
through the book. Write and solve an
equation to find the number of pages in
the book.
7.
𝑦−1
4
𝑚
7
−3
= −2
8. -x – 4 = 9
9. The junior class is selling ganola bars to
raise money. They purchase 1250
granola bars and paid a delivery fee of
$25. The total cost, including the delivery
fee, was $800. What was the cost of
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each granola bar?
Do You Know How?
Solve each equation. Check your answer.
Solving mult-step equations
Solving equations with variables on
both sides
10. 7p + 8p – 12 = 59
15. 3x + 4 = 5x – 10
11. -2(3x + 9) = 24
16. 5(y – 4) = 7(2y + 1)
12.
2𝑚
7
+
3𝑚
14
=1
17. 2a + 3 =
1
2
6 + 4𝑎
13. 1.2 = 2.4 – 0.6x
18. 4x – 5 = 2(2x + 1)
14. There is a 12-ft fence on one side of a
rectangular garden. The gardener has
44 ft of fencing to enclose the other
three sides. What is the length of the
garden’s longer dimension? (draw a
19. Pristine Printing will print business cards
for $0.10 each plus a setup charge of
$15. The Printing Place offers business
cards for $0.15 each with a setup
charge of $10. What number of business
cards costs the same from either
printer?
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diagram, if it helps)
Do You UNDERSTAND?
Solving one-step equations
Which property of equality would you use to
solve each equation? Why? See pg. 3-4
20. 3 + x = -34
21. 2x = 5
22. x – 4 = 9
Solving two-step equations
What properties of equality would you use to
solve each equation? What operation you
perform first? Explain
𝑠
4
25. -8 = + 3
26. 2x – 9 = 7
𝑥
3
23. 𝑥/7=9
27.
24. Write a one-step equation. Then write
two equations that are equivalent to
your equation. How can you prove that
all three equations are equivalent?
28. -4x + 3 = -5
−8=4
𝑑−3
29. Can you solve the equation
= 6 by
5
adding 3 before multiplying by 5?
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Explain
Do You UNDERSTAND?
Solving multi-step equations
Identities and Equations with no
Solutions
Explain how you would solve each equation. Match each equation with the appropriate
number solutions.
30. 1.3 + 0.5x = -3.41
34. 3y – 5 = y + 2y – 9
A. infinitely many
31. 7(3x – 4) = 49
35. 2y + 4 = 2(y + 2)
B. one solution
2
7
32. − 9 𝑥 − 4 = 18
36. 2y – 4 = 3y – 5
C. no solution
33. Ben solves the equation -24 = 5(g + 3) by
37. A student solved an equation and found
first dividing each side by 5. Amelia
that the variable was eliminated in the
solves the equation by using the
process of solving the equation. How would
Distributive Property. Whose method do
the student know whether the equation is
you prefer? Explain
an identity or an equation with no solution?
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